Journal of Glaciology, Vol. 53, No. 182, 2007
363
The effect of bottom boundary conditions in the ice-sheet to
ice-shelf transition zone problem
Alexander V. WILCHINSKY
Centre for Polar Observation and Modelling, Department of Earth Sciences, University College London, Gower Street,
London WC1E 6BT, UK
E-mail: [email protected]
ABSTRACT. Uniqueness of the transition zone solution is discussed. It is argued that when it is assumed
that the stresses are continuous at the grounding line, the ice-shelf solution at the grounding line should
possess a zero slope. In order to avoid issues caused by the mathematical singularity of the solution, the
same technique as that used to study the transition zone is applied to a similar problem of lifting an
elastic sheet from a rigid substrate, which allows a better physical understanding. This exemplifies the
effect of the bottom boundary condition of no vertical motion imposed up to a fixed grounding line
position, while the forcing parameters, such as the flow rate, vary. Its effect is to produce multiple
solutions due to suppression of the separation mechanism.
1. INTRODUCTION
Ice-sheet motion is determined by the transition zone
dynamics that couple the grounded part of an ice sheet to
its floating part, the ice shelf. As the shallow-ice approximation makes equations governing the grounding part of the ice
sheet simpler, it is convenient to decouple the grounded icesheet problem from the ice-shelf problem. This leads to the
necessity of imposing boundary conditions at the grounding
line. Although it has been argued that, due to its low stress,
the presence of the ice shelf does not constrain the ice-sheet
dynamics in any way apart from the condition of hydrostatic
equilibrium at the grounding line (Hindmarsh, 1993), here
we discuss the other possibility, namely that the ice-sheet
motion is uniquely determined by its external and initial
conditions. The latter implies two boundary conditions at
the grounding line that, together with the condition of zero
mass flux at the ice divide, close the isothermal ice-sheet
problem described by a non-linear parabolic second-order
differential equation solved between the ice divide and the
unknown grounding line position.
To date, no work has been able to show uniqueness of the
transition zone solution that, apart from the hydrostatic
equilibrium condition, would allow us to impose an
additional unique boundary condition at the grounding line
to relate the ice thickness to the mass flow rate. The progress
in doing so has been hindered by the complexity of the
problem, especially due to a singularity of the solution at the
grounding line caused by a jump in the bottom boundary
conditions.
Chugunov and Wilchinsky (1996) used a sigma transformation of the coordinate system and finite differences to
solve the ice-sheet to ice-shelf transition problem in its
steady-state form. However, in order to find a constant that,
firstly, for non-linear rheology with n being the Glen’s flowlaw exponent, can be written as (Wilchinsky and Chugunov,
2001)
ðnþ2Þ=n . 1=n
/ h xg
q xg
ð1Þ
and, secondly, appears when both the mass flow rate, q, and
the thickness, h, at the grounding line (x = xg) are
normalized to unity in the transition zone, they had to
assume that the sought ice-shelf bottom is horizontal at the
grounding line. The resulting magnitude of then determines the sought, unique relationship between the ice
thickness and flow rate at the grounding line that constitutes
the second boundary condition. The necessity to make an
assumption that the ice-shelf bottom is horizontal at the
grounding line was justified neither physically nor mathematically. Without this assumption the solution of the
transition problem could be generally found for an arbitrary
ice-shelf bottom slope and a fixed ice thickness at the
grounding line, that could lead to the conclusion that there
is no unique connection between the mass flow rate and the
ice thickness at the grounding line.
Although the problem of uniqueness of the transition
zone solution could be solved eventually using rigorous
mathematical analysis, here we aim at finding some insights
into the issue that can be drawn through an intuitive
understanding. In particular, we consider the similar problem of lifting an elastic sheet with an unknown lift line that,
when solved using the same technique as the transition zone
problem, produces similar results but allows a better
physical understanding. Our purpose here is to outline a
way which more detailed studies can follow in order to find
the final answer to the problem, as any, even small, progress
in understanding this contentious problem should be made
available to ice-sheet modellers for further discussion. In
particular, here we argue that the possibility of multiple
solutions of the transition zone problem may be due to the
specific way of posing the mathematical problem at the
grounding line that prevents separation of the ice from the
bedrock. We also show that if the physical interaction
between the grounded ice and the water in the vicinity of the
grounding line is taken into account, then a unique
relationship between the mass flow rate and the ice
thickness at the grounding line should exist.
2. THE GROUNDING LINE POSITION
The problem of marine ice-sheet dynamics requires different
boundary conditions at the bottom of the grounded part and
the floating part separated by the grounding line, xg (Fig. 1).
We consider two-dimensional flow of an ice sheet. At the
grounded ice-sheet bottom (z = b, where b is the ice-shelf
bottom elevation; x < xg) that is considered here to be flat
364
Wilchinsky: Bottom boundary conditions in the ice-sheet to ice-shelf transition
Fig. 1. Schematic of a marine ice sheet.
and horizontal for simplicity, a sliding law is applied
together with an assumption of negligible melting
m
vx ¼ kxz
, x < xg ,
ð2Þ
vz ¼ 0, x < xg ,
ð3Þ
where (x,z) are the horizontal and vertical coordinates, vx
and vz are horizontal and vertical velocities, xz is the
bottom shear stress, k is a friction coefficient and m is an
exponent. The condition k = 0 yields a no-slip condition at
the bottom. Some significant sliding can be present due to
complex basal processes in the transition zone, and the noslip condition is used purely for mathematical simplicity, in
order to represent the problem in its most basic form
(Lestringant, 1994; Chugunov and Wilchinsky, 1996).
At the ice-shelf bottom, stress continuity yields
db
xz zz ¼ hw w g,
dx
ð4Þ
db
db
ð5Þ
xx ¼
hw w g, x > xg ,
dx
dx
where b is the ice-shelf bottom elevation, hw is the water
depth, w is the water density, g is gravity acceleration and
zz and xz are the corresponding components of the full
stress tensor. The grounding line position, xg, is not a
prescribed magnitude, but is determined by the dynamics of
the lower surface of the ice. In particular, if the ice-shelf
bottom sinks up to the bedrock, the grounding line
advances. If the sea water can penetrate and lift the
grounded ice in the vicinity of the grounding line, the
grounding line retreats. From this it is clear that the ice sheet
stays grounded when the water pressure, hwwg, is smaller
than the stress with which the ice pushes downwards along
the normal, n, to its surface, n n, so that the sea water
cannot penetrate under the ice sheet:
xz n ðbÞ n ¼ zz ðbÞ > hw w g, x < xg :
ð6Þ
At the same time at the ice-shelf bottom these stresses are
equal:
n ðbÞ n ¼ hw w g, x > xg ,
ð7Þ
that is, effectively Equation (4). Therefore in general terms
the grounding line position is effectively determined by two
conditions at the ice bottom
n ðbÞ n ¼ hw w g, x ¼ xg þ,
ð8Þ
½n ðbÞ nx¼xg > 0, x ¼ xg ,
ð9Þ
where the brackets denote a jump while the þ and signs
denote the limits from the right and left to Xg respectively. If
we now abstract, for a moment, from the issues associated
Fig. 2. Different cases of the ice-shelf bottom elevation, b, for a
fixed ice thickness at the grounding line: (a) a zero slope at the
grounding line is determined by the integral mass flow rate, q,
equal to q0; (b) a positive slope is determined by q < q0; and (c) a
negative slope is determined by q > q0.
with the singularity of the mathematical problem, and make
the realistic physical assumption that the stress is continuous
across the grounding line in real ice sheets, then Equations (8) and (4) yield
db
¼ 0, x ¼ xg ,
dx
ð10Þ
so the shelf slope at the grounding line is horizontal.
It should be noted that conditions (8) and (9) or (10)
require knowledge of the stress or the shelf bottom profile in
the transition zone. Therefore they are not convenient to use
in the ice-sheet models written in terms of the shallow-ice
approximation. One therefore strives to determine conditions for the mass flow rate and the ice thickness at the
grounding line that could determine the grounding line
position. In order to do this the ice flow in the ice-sheet to
ice-shelf transition zone must be considered.
3. THE TRANSITION ZONE PROBLEM
Here we summarize main features of the transition zone
solution that were described by Wilchinsky and Chugunov
(2001), where more details can be found. A no-slip
condition was used at the bedrock, and a perturbation
analysis allows us to reduce the transition zone problem to
solution of the full Stokes equations in a domain with a flat
upper surface coinciding with the sea level to order
¼ ðw i Þ=w 0:1, where i is the density of ice
(Fig. 2). The sea bottom was approximated to be flat, while
the ice-shelf bottom is an unknown surface. A special nondimensionalization ensured that the mass flow rate and the
grounding line thickness are both unity, which leads to the
appearance of the parameter in the shelf thickness
equation. The grounding line position is fixed. Variation of
in this case is analogous to variation of the mass flow rate,
q(xg), when the ice thickness (or water depth) is fixed.
Because of the fixed grounding line position, such a
variation of is not allowed to affect the grounding line
position. Because of this, the only way to determine whether
a solution of the transition zone problem with a fixed
grounding line is realistic is to check whether the conditions
determining the grounding line, (8) and (9) or (10), hold.
Typical ice-shelf bottom profiles for different mass flow
rate values, q(xg), are presented schematically in Figure 2,
Wilchinsky: Bottom boundary conditions in the ice-sheet to ice-shelf transition
365
generated during the numerical analysis described above.
This effect of the variation in mass flow rate is supported by
recent finite-element simulations (S. Nowicki and
D. Wingham, unpublished information). It can also be
inferred using much simpler arguments. If we consider a
constant flow rate solution of an ice shelf of unit thickness
at the grounding line, found using the shallow-ice
approximation (MacAyeal and Barcilon, 1988)
h ¼ ½1 þ ð þ 1Þx=q1=ðnþ1Þ ,
ð11Þ
where x is the scaled distance from the grounding line,
then h0 ð0Þ ¼ 1=q. The limits of the idealized ice-shelf
solution, h0 ð0Þ ! 0 as q ! 1 and h0 ð0Þ ! 1 as q ! 0,
reflect the general behaviour of such a transition zone
solution: when the ice thickness is fixed, as the mass flow
rate decreases the bottom slope becomes more positive;
and as the mass flow rate increases the ice-shelf bottom
slope becomes more negative (Fig. 2). Furthermore, within
the transition zone, the effect of the no-slip condition used
in its grounded part is still significant, so the horizontal
velocity increases with distance from the bottom. This
implies that the shear stress in the ice shelf is non-negative,
and Equation (4) shows that the stress, –zz, acting on a
horizontal plane from above is larger than the hydrostatic
stress when db/dx < 0 (negative bottom slope) and smaller
than the hydrostatic stress when db/dx > 0 (positive bottom
slope). The situation with a negative bottom slope clearly
shows that the ice sheet would advance.
The simple condition for the shelf bottom slope Equation (10) was derived here assuming continuity of the
stresses at the grounding line. It was also used by
Chugunov and Wilchinsky (1996) in order to determine in Equation (1), which provides the sought relationship
between the mass flow rate and the ice thickness at the
grounding line. The latter was done without any justification, and it is now clear that it is tantamount to requiring
that the stress is continuous at the grounding line. A unique
makes it possible to identify different grounding line
positions, for example, by varying the water depth (Fig. 3).
However, the assumption of continuity can be questioned,
as a problem describing a jump between a no-slip
condition and a slip condition yields infinite stresses
(Barcilon and MacAyeal, 1993). Without this assumption
one has to use more general conditions (Equations (8) and
(9)). Because of the singularity at the grounding line, these
conditions may be satisfied for a range of , determining a
range of non-zero angles of the shelf bottom slope
(S. Nowicki and D. Wingham, unpublished information).
In this case multiple solutions may exist. One should,
however, take into consideration that the described jump in
the boundary condition makes the continuum mechanical
model invalid at the jump itself where the strain rates are
discontinuous and stresses are infinite. Because of this, it is
not clear if one can safely reach any conclusion about
uniqueness of the grounding line position in real ice sheets
by studying the singular mathematical solution. A no-slip/
slip singularity on a flat surface is similar to a singularity at
the contact line of a drop moving with no slip over a
surface. The latter singularity is usually avoided by adopting
a slip condition (Equation (2)) near the contact line
(Hocking, 1976). Similarly, application of a sliding boundary condition instead of a no-slip condition would remove
the singularity, provided that the bottom slope is continuous at the grounding line. If it is not, then the condition
Fig. 3. Different grounding line positions determined by different
water depths. The cumulative mass flow rate increases with x due to
a constant ice accumulation rate at the ice-sheet surface.
of no vertical motion at the bottom of the grounded ice
sheet (Equation (3)) would still give rise to a stress
singularity at the grounding line. In order to avoid this
complex mathematical issue, we will refer to a problem of
elasticity that, firstly, has similar characteristics, and,
secondly, allows a much easier experimental understanding
of the uniqueness problem.
4. LIFTING AN ELASTIC SHEET FROM A SUBSTRATE
Our aim here is to study the effect of condition (3) imposed
up to a fixed grounding line position on the solution around
the grounding line while other parameters determining
external forcing, like the mass flow rate, vary. Let us first note
that the rheology of an incompressible linearly elastic
material is described as
E
@ui @uj
ij ¼
,
ð12Þ
þ
2ð1 þ Þ @xj @xi
where ui and uj are displacements, E is Young’s modulus and
is Poisson’s ratio. This formula is similar to that which
describes the rheology of an incompressible linear fluid
when one uses velocities, v, instead of displacements, u. The
no-slip condition can then be given as
u ¼ 0, z ¼ b,
ð13Þ
and the mass balance as
r u ¼ 0:
ð14Þ
Therefore, if we consider incompressible elastic deformation
under a body force acting along x of an elastic material
confined between two horizontal planes with adhesion over
the ‘upstream’ part of the bottom plane and frictionless slip
along the rest of the planes, we would derive the same set of
equations as those for a linear fluid flow problem with a noslip/slip transition problem, but in terms of displacement, u.
This implies the same type of singularity of the stress at a
jump between adhesion and free slip when an incompressible elastic material is considered.
Let us now consider the problem of lifting an elastic sheet
(e.g. rubber) from a rigid substrate (Fig. 4). One edge of the
sheet is fixed, while the other is lifted with a force, F, against
gravity. We assume no friction at the substrate. The lifting
force causes vertical displacement that depends on the force
magnitude. The length of the lifted part will be longer for
larger lifting forces and shorter for smaller lifting forces. Such
behaviour is similar to the ice-sheet length change due to a
change in, say, water depth (Fig. 3).
366
Wilchinsky: Bottom boundary conditions in the ice-sheet to ice-shelf transition
Fig. 4. Schematic of flexure of an elastic sheet on a rigid substrate
under a lifting force acting at its edge: (a) a force, F = F0; (b) a larger
force, F > F0; (c) a smaller force, F < F0.
We do not aim to find the exact mathematical solution of
the problem, but rather appeal to physical experience that
could be corroborated by future experimental studies. We
apply the same approach as that used in transition zone
studies to understand its effect at a physical level.
If we now want to find the position of the lift line,
denoted here by xg, for some force, F0, and follow the
standard method used in modelling the ice-sheet transition
zone, then we impose the condition of no vertical motion,
Equation (3), up to (as yet unknown) xg, preventing the
bottom of the sheet from separating from the substrate, while
the rest of the bottom is considered to be a free surface. If
now the lifting force is varied we can argue, from physical
experience, that the condition of no vertical motion,
Equation (3), imposed up to the fixed xg provides multiple
solutions of the problem. These solutions are schematically
drafted in Figure 5, where the lower surface profiles remind
us of those of the ice-shelf solutions in the transition zone
shown in Figure 2. Evidently, only one solution presented in
Figure 5 corresponds to the natural solutions shown in
Figure 4, while the others are artefacts of using Equation (3)
within a fixed region (that does not allow the sheet to come
off) as well as not considering the full length of the solid
substrate (that would not allow the sheet to droop). In these
real situations the stress is continuous and the correct
solution can be identified from Figure 5 by inspecting the
lower surface angle of 08. The normal stress acting on the
substrate is zero at the lift line. The other solutions are
described by finite lift angles to ensure the bottom normal
stress is finite at the lift line. This is also supported by the
theory of peeling at finite local peel angles (Kinloch and
Williams, 1994). The presence of a finite angle may lead to a
mathematical stress singularity in elastic materials (Kotousov
and Lew, 2006), the detailed examination of which is
beyond the scope of this work. An important conclusion,
however, is that the possible presence of such a mathematical singularity does not change the fact that the only
solution of the problem shown in Figure 5 that is a solution
of the original problem, shown in Figure 4, is the one
possessing a zero lower surface angle.
5. CONCLUSIONS
Our conclusion is that the main difficulty in rigorously
treating the uniqueness problem of the transition zone
solution arises from a mathematical singularity at the
grounding line. If continuity is assumed, then a unique
relationship between the ice thickness and the integral
mass flow rate at the grounding line may be found, if one
takes into account the bottom stress while using bottom
Fig. 5. Schematic of flexure of an elastic sheet on a rigid substrate
with no vertical displacement imposed at the bottom, under a lifting
force acting at its edge: (a) a force, F = F0; (b) a larger force, F > F0;
(c) a smaller force, F < F0. The solid substrate edge is given by the
separation point, xg, determined by application of the force F ¼ F0 .
boundary conditions that prevent vertical motion. This is
tantamount to requiring that the sought solution has a zero
bottom slope at the grounding line in the floating part. The
reason that multiple solutions of the problem are found is
that imposing idealized boundary conditions that prevent
vertical motion implies the application of an additional
force, pulling the grounded ice sheet to the bedrock in the
region where the grounded ice sheet is assumed to rest on
the bedrock, whatever the other circumstances are. This
prevents the ice from naturally retreating or advancing, and
allows different mathematical solutions of the transition
zone problem; in particular, mathematical solutions with
the same thickness but different flow rates can exist. In
reality such ice sheets would not be stationary, as the water
pressure would be enough for sea water to penetrate
beneath the ice, lift the grounded ice sheet and make it
retreat. Therefore, in order to choose the physically relevant
solution, the vertical stress must be considered at the
bottom of the grounded ice sheet and compared to the
water pressure. The only situation in this case when there
would be no tendency for the ice sheet to either advance or
retreat is when the water pressure is equal to the vertical
normal stress in the grounded ice sheet at the grounding
line. As the grounded ice-sheet bottom is assumed to be
horizontal in such simplified studies, with a physical
assumption of continuity of the stresses across the grounding line, this is tantamount to requiring the ice-shelf bottom
slope to be horizontal at the grounding line (Fig. 2a). With
this condition a unique relationship between the ice
thickness and the mass flow rate at the grounding line
may be found.
REFERENCES
Barcilon, V. and D.R. MacAyeal. 1993. Steady flow of a viscous ice
stream across a no-slip/free-slip transition at the bed. J. Glaciol.,
39(131), 167–185.
Chugunov, V.A. and A.V. Wilchinsky. 1996. Modelling of a marine
glacier and ice-sheet–ice-shelf transition zone based on asymptotic analysis. Ann. Glaciol., 23, 59–67.
Hindmarsh, R.C.A. 1993. Qualitative dynamics of marine ice
sheets. In Peltier, W.R., ed. Ice in the climate system. Berlin, etc.,
Springer-Verlag, 67–99.
Hocking, L.M. 1976. A moving fluid interface on a rough surface.
J. Fluid Mech., 76(4), 801–817.
Kinloch, A.J., C.C. Lau and J.G. Williams. 1994. The peeling of
flexible laminates. Int. J. Frac., 66(1), 45–70.
Kotousov, A. and Y.T. Lew. 2006. Stress singularities resulting from
various boundary conditions in angular corners of plates of
Wilchinsky: Bottom boundary conditions in the ice-sheet to ice-shelf transition
arbitrary thickness in extension. Int. J. Solids Struct., 43(17),
5100–5109.
Lestringant, R. 1994. A two-dimensional finite-element study of
flow in the transition zone between an ice sheet and an ice shelf.
Ann. Glaciol., 20, 67–72.
367
MacAyeal, D.R. and V. Barcilon. 1988. Ice-shelf response to icestream discharge fluctuations: I. Unconfined ice tongues.
J. Glaciol., 34(116), 121–127.
Wilchinsky, A.V. and V.A. Chugunov. 2001. Modelling ice flow in
various glacier zones. J. Appl. Math. Mech., 65(3), 479–493.
MS received 18 August 2006 and accepted in revised form 24 February 2007